If I happen to say to someone, Tom visits him more than me, what exactly do I mean? That Tom takes the time to see him more often than I do, or that Tom visits him more often than Tom visits me? What is at issue here is the grammatical technicality called case, which it can be fashionable to malign but advantageous to recognize.
The derivation of the word case is a bit odd, so we can’t rely on that to frame the concept. We can say, though, that case refers to the spelling or position of a word that shows its grammatical function in a sentence. There are three cases in English with three cumbersome names (nominative, possessive, and objective), and the phenomenon occurs in nouns and pronouns. It’s easiest to see what’s going on by looking at the pronouns.
We know that he, his, and him all have to do with that person over there, and I, my, me with the person writing here, but why three forms to refer to one thing? The three different spellings mark the three cases, and each case is assigned certain grammatical jobs. The nominative case (he, I) marks the subject of a clause (when he talks, I listen), the possessive case (his, my) shows the possessor of something (his remarks, my attention), and the objective case (him, me) identifies the object of a verb or preposition (they invited him instead of me).
With this little bit of grammatical knowledge, we can sort out the meaning of ambiguous statements very quickly. In our original sentence, Tom visits him more than me, the two pronouns (him and me) are in the objective case because they denote two objects of the verb visits; Tom visits two people, him and me, and he visits him, unfortunately, more often than he visits me. There can be no question of my visiting anyone because I do not appear in the nominative case, the subject of any action.
As is often the case in the study of grammar (no pun intended), what is really a rich and subtle affair is too easily dismissed as pedantry. It is certainly correct to observe that modern English does not always conform strictly to these case rules, particularly in more informal settings: If he wants to get rid of him, he’s going to fire him, not me. In this emotive statement, the purist would demand I, not me, because a second subject is being indicated and therefore the nominative case is required. The pure, though, are indeed often unvisited. Most of us would be loath to say or write not I, because a too strict interpretation of the grammatical law would violate the fiery spirit of the circumstance and put us outside the living moment.
And yet we need the laws like rivers need their banks or any game its rules—especially when we run the risk of sacrificing logic and clarity. As our first sentence illustrates, change the case and you can change the very meaning of a statement. This tension between when to follow and when to violate the rules is fundamental to the art of writing (and to any other art) because art arises from life, an ongoing scene of wrong and right, dark and light. Awareness alone can decide.